The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics

摘要

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S1xS3. This obstruction is given in terms of the relation between the Froyshov invariant of the generator of H3(X;Z) with the 4-dimensional Casson invariant lambda SW(X) defined in [Mrowka, Ruberman and Saveliev, J. Differential Geom. 88 (2011) 333-377]. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.